Starting procedure of open-loop vector control in synchronous machine

ABSTRACT

A method for starting an open-loop vector control of a synchronous machine by determining a stator inductance model of the machine; measuring the stator iductance in a plurality of directions; arranging the measured inductances in the model to form model parameters resulting in minimum error; checking the polarity of rotor magnetization to verify the direction thereof; initializing flux linkages of the open-loop vector control according to the model parameters and the direction of rotor magnetization; and starting the machine by the open-loop vector control method.

BACKGROUND OF THE INVENTION

The present invention relates to a method of starting open-loop vectorcontrol in a synchronous machine, the method comprising the steps ofdetermining a stator inductance model of the synchronous machine andmeasuring the stator inductance in a plurality of directions.

Vector control refers to a manner of controlling an AC motor whichallows flux linkage and torque of the motor to be controlledindependently like in a DC motor. In the DC motor, direct currentsinfluencing the flux linkage and the torque are controlled, while in theAC motor both the amplitude and the phase angle of the currents have tobe controlled. Thus, current vectors are controlled, from which comesthe term vector control.

To implement vector control, the flux linkage and the current of themotor have to be known. The flux linkage of the motor is generated bythe action of stator and rotor currents in the inductances of themachine. In an asynchronous machine, the rotor current has to beestimated and the estimation requires information on rotation speed ofthe rotor. This requires measured or estimated rotation speed of therotor. In the synchronous machine, a magnetization current independentof stator magnetization is applied to the rotor, or the rotormagnetization is implemented with permanent magnets and its influence,seen from the stator, shows in the direction of the rotor positionangle. To know the flux linkage caused by the position angle, theposition angle of the rotor has to be measured or estimated.

When vector control of the AC motor employs a measured rotation speed orposition angle of the rotor, the control method is known as theclosed-loop vector control. If the rotation speed or the position angleis estimated, the control method is known as the open-loop vectorcontrol. Depending on the implementation method, a variable to beestimated can also be the stator flux linkage, apart from the rotorangle or angular speed.

When the synchronous machine is started by vector control, the machine'sstator flux linkage has the initial value ψ_(s0), which is dependent onthe rotor magnetization ψ_(f) and the rotor position angle θ_(r) asfollows:

ψ_(s0)=ψ_(f) e ^(jθ) ^(_(r)) .

When voltage u_(s) influences the stator flux linkage, the stator fluxlinkage changes in accordance with the equationψ_(s) = ψ_(s0) + ∫_(t₁)^(t₂)(u_(s) − R_(s)i_(s))t.

It appears from the equation that when integrating the stator fluxlinkage a previous value of the stator flux linkage is required, apartfrom the voltage and current values. Thus, to start the machinecontrollably, information on the initial position angle of the rotor isrequired. When employing the closed-loop vector control, the initialangle is measured, whereas when employing the open-loop vector control,the initial angle has to be defined by estimation. When the rotorrotates, the rotor flux linkage generates an electromotive force whichcan be utilized in vector control in a normal operating situation, butat a rotor standstill there is no electromotive force.

In a salient pole synchronous machine, such as a separately excitedsynchronous machine or one with permanent magnet magnetization or in asynchronous reluctance machine, the stator inductance L_(s) instationary coordinates varies as a function of the rotor angle θ_(r) aspresented in the following equation

L _(s) =L _(s0) +L _(s2) cos2 θ_(r),

and FIG. 1 shows a graphic illustration of the equation. It appears fromthe figure that the inductance varies around the basic value L_(s0) attwice the rotor angle in a magnitude indicated by the inductancecoefficient L_(s2). The inductance coefficients L_(s0) and L_(s2) aredefined as follows${L_{s0} = \frac{L_{sd} + L_{sq}}{2}},{L_{s2} = \frac{L_{sd} - L_{sq}}{2}},$

where the inductances L_(s0) and L_(sq) are the direct-axis andquadrature-axis transient inductances of the synchronous machine.

To utilize the above equation for defining the initial angle of therotor is known per se and it is set forth, for instance, in the articlesby S. Östlund and M. Brokemper “Sensorless rotor-position detection fromzero to rated speed for an integrated PM synchronous motor drive”, IEEETransactions on Industry Applications, vol. 32, pp. 1158-1165,September/October 1996, and by M. Schroedl “Operation of the permanentmagnet synchronous machine without a mechanical sensor”, Int. Conf. OnPower Electronics and Variable Speed Drives, pp. 51-55, 1990.

According to the article by M. Leksell, L. Harnefors and H.-P. Nee“Machine design considerations for sensorless control of PM motors”,Proceedings of the International Conference on Electrical MachinesICEM'98, pp. 619-624,1998, sinusoidally altering voltage is supplied toa stator in the assumed direct-axis direction of the rotor. If thisresults in a quadrature-axis current in the assumed rotor coordinates,the assumed rotor coordinates are corrected such that thequadrature-axis current disappears. The reference states that aswitching frequency of the frequency converter supplying the synchronousmachine should be at least ten times the frequency of supply voltage.Thus, the supply voltage maximum frequency of a frequency convertercapable of 5 to 10 kHz switching frequency, for instance, is between 500and 1000 Hz. This is sufficient for an algorithm to function. Switchingfrequencies as high as this are achieved by IGBT frequency converters,but frequency converters with GTO or IGCT power switches, required athigher powers, have the maximum switching frequency of less than 1 kHz.The maximum frequency of the supply voltage in the initial angleestimation remains then below 100 Hz. At such a low frequency themachine develops torque and the algorithm becomes considerably lessaccurate.

In the reference by M. Schroedl, 1990, the initial angle is calculateddirectly from one inductance measurement, or, if more measurements areemployed, the additional information is utilized by eliminating theinductance parameters. A drawback with the method is that an error,which is inevitable in measuring, has a great influence. One example ofactual inductance measurement with a permanent magnent machine at rotorangles θ_(r)=[0, . . . ,2π] is shown in FIG. 2. The figure showstheoretically great deviations from the sine curve. The inductancemeasurement is effected such that a stator is fed with a current impulsewhich causes flux linkage on the basis of which the inductance iscalculated. Errors may arise from an error in current measurement orfrom the fact that the measuring current produces torque that swings therotor.

From the inductance expression in the stationary coordinates it ispossible to derive an expression for a rotor angle${\theta_{r} = {{\frac{1}{2}\arccos \frac{L_{s} - L_{s0}}{L_{s2}}} + {n\quad \pi}}},$

where n is an integer. The influence of the error in the measured L_(s)can be studied by differentiating θ_(r) with respect to L_(s):$\frac{\theta_{r}}{L_{s}} = {{- \frac{1}{2L_{s2}}}\quad {\frac{1}{\sqrt{1 - \left( \frac{L_{s} - L_{s0}}{L_{s2}} \right)^{2}}}.}}$

This allows calculating an error estimate for the angle${\Delta \quad \theta_{r}} \approx {\frac{\theta_{r}}{L_{s}}\Delta \quad {L_{s}.}}$

It is observed that$\left. {\Delta \quad \theta_{r}}\rightarrow\infty \right.,\quad \left. {{when}\quad \frac{L_{s} - L_{s0}}{L_{s2}}}\rightarrow 1. \right.$

It is observed from the above that, when the inductance differencebetween the direct-axis and quadrature-axis directions is small, theerror estimate of the angle approaches infinite. In other words, initialrotor angle definition based on inductance measurings becomes the moreunreliable, the closer to one another the magnitudes of the direct-axisand quadrature-axis inductances of the rotor.

In the method presented in the reference by S. Östlund and M. Brokemperthe rotor angle is not calculated directly, but the minimum inductanceis searched by starting the measuring of inductances in differentdirections first at long intervals and when approaching the minimum byreducing the angular difference of successive measurings. Even though itis not mentioned in the article, the method easily catches fictitiousminima resulting from measuring errors, and therefore, an error valuemay be extremely high.

On the basis of the above, the influence of the inductance measuringerrors should be reduced somehow. One method could be to employ severalmeasurings in each direction and to calculate the average from themeasured inductances, yet this procedure does not eliminate theinfluence of a systematic error.

BRIEF DESCRIPTION OF THE INVENTION

The object of the present invention is to provide a method which avoidsthe above-mentioned disadvantages and enables starting of an open-loopvector control in a synchronous machine in a reliable manner. This isachieved by the method of the invention which is characterized bycomprising the steps of arranging measured stator inductances in adetermined stator inductance model in order to form model parametersgiving the minimum error, checking magnetization polarity of a rotor inorder to verify the direction of the rotor magnetization, initializingflux linkages of the open-loop vector control according to the formedmodel parameters and the direction of the rotor magnetization, andstarting the synchronous machine by the vector control method.

The method of the invention is based on the idea that the magnitude ofthe stator inductance is measured in a plurality of directions and theinductance values obtained as measurement results are arranged in theinductance model of the machine. As a result of the arrangement, veryaccurate information on the initial rotor angle of the synchronousmachine is obtained. In addition, by utilizing the method of theinvention, information on the initial value of the rotor magnetizationin the stationary coordinates is obtained, and consequently the machinecan be started in a reliable manner without transients or jerkingstartup.

BRIEF DESCRIPTION OF THE DRAWINGS

In the following the invention will be described in greater detail inconnection with preferred embodiments with reference to the attacheddrawings, wherein

FIG. 1 is a graph of a stator inductance L_(s) as a function of anangle;

FIG. 2 shows measured magnitudes of stator inductance as a function ofan angle;

FIG. 3 shows the relations between the coordinates;

FIG. 4 shows the order of stator inductance measurings; and

FIG. 5 is a flow chart of the starting procedure in accordance with theinvention.

DETAILED DESCRIPTION OF THE INVENTION

According to the invention, starting of an open-loop vector control in asynchronous machine first comprises the step of determining a statorinductance model of the synchronous machine L_(s)=L_(s0)+L_(s2) cos2θ_(r). The equation shows how the inductance depends on the rotor anglein stationary coordinates. The equation thus proves how the inductancemeasured in the direction of x-axis of the stator coordinates changeswhen the rotor is rotated a degree of an angle θ_(r).

When it is desired to determine the initial rotor angle, it is notpossible to rotate the rotor at different rotor position angles formeasuring the inductance. Instead, the stator coordinates are rotated tothe angle, in the direction of which the inductance is measured. Theserotatable stator coordinates are referred to as virtual statorcoordinates. FIG. 3 illustrates various coordinates and they areindicated such that the stator coordinates are coordinates xy, thevirtual stator coordinates are coordinates x′y′ and the rotorcoordinates are dq.

Then the inductance model in the virtual stator coordinates may bewritten as $\begin{matrix}{L_{s} = \quad {L_{s0} + {L_{s2}{\cos \quad\left\lbrack {2\left( {\theta_{r} - \lambda} \right)} \right\rbrack}}}} \\{= \quad {L_{s0} + {L_{s2}{\cos \quad\left\lbrack {2\left( {\lambda - \theta_{r}} \right)} \right\rbrack}}}} \\{{= \quad {L_{s0} + {L_{s2}\cos \quad \left( {{2\lambda} + \phi} \right)}}},}\end{matrix}$

where θ_(r) is the rotor angle in the stationary coordinates, λ is anangle in the virtual coordinates and θ_(r)−λ is the rotor angle in thevirtual coordinates. By introducing a parameter ψ=−2θ_(r), it ispossible to simplify the equations under study. So the desired rotorangle in the stationary coordinates is given by$\theta_{r} = {- {\frac{\phi}{2}.}}$

With reference to the flow chart of FIG. 5, in accordance with theinvention, after initializing 2 the inductance measuring the statorinductance of the synchronous machine is measured 3 in a plurality ofdirections. The stator inductance measurement is advantageously effectedsuch that the stator is fed with a voltage pulse generating a currentpulse which causes flux linkage on the basis of which the inductance canbe calculated. The stator inductance measurement can be performed, forinstance, in six directions. These six voltage vectors can readily beimplemented with inverters, since the vectors correspond to inverterswitch combinations, in which a positive and a negative voltage isgenerated between each two poles of the three-phase stator. FIG. 4illustrates six current vectors and their preferable mutual order inconnection with measuring. For improved accuracy, a plurality ofmeasurings can be performed in each direction. In order to reduce thepossibility that the rotor turns, the measurings are performed in theorder shown in FIG. 4, i.e. measuring starts in the direction of 0°,thereafter in the direction of 180°, thereafter in the direction of 60°and so on, until all the directions are measured. FIG. 2 shows measuredvalues of the stator inductance with different rotor angle values. Inthe flow chart of FIG. 5, it is checked 4 after the inductancemeasurement 3 if the inductances of all directions determined to bemeasured are measured, and if not, a new direction is initialized 5 formeasuring the inductance in accordance with FIG. 4.

According to the invention, the measured stator inductances are arranged6 in the determined stator inductance model in order to form modelparameters giving the minimum error. Measurement data is preferablyarranged in the model by using the method of least mean squares (LMS).In the LMS method, a model is formed for a measurable variable, in whichmodel the data to be measured is arranged such that the square sum isminimized. The square sum refers to a sum of squares of the differenceof the measured values and corresponding model values. If the model islinear, the parameters of the model can be solved in a closed form, butin a nonlinear case the question is about a numerically solvablenonlinear optimization task.

In a nonlinear LMS method the following function is minimized${F(a)} = {{\sum\limits_{i = 1}^{m}\left\lbrack {f_{i}(a)} \right\rbrack^{2}} = {{\sum\limits_{i = 1}^{m}{{f_{i}(a)}{f_{i}(a)}}} = {{\langle{{f(a)},{f(a)}}\rangle} = \left\lbrack {f(a)} \right\rbrack^{2}}}}$

where

m is the number of samples (measurements),

ƒ_(i)=y_(i)−M(t_(i),a)is the difference of a nonlinear model M and themeasured data (t_(i),y_(i)),

a=[a₁ a₂ a₃ . . . a_(n)]^(T) ε^(n) are the model parameters,

f(a)=[ƒ₁(a) ƒ₂(a) . . . ƒ_(m)(a)]^(T) is a vector formed by functionsƒ_(n)

(,) refers to an inner product.

A gradient of the function F(a) needed in a numerical solution is

∇F(a)=2J(a)^(T) f(a)

where J(a) is the Jacobian matrix of the function f(a).

In the method, the inductance model is indicated by M and its parametersby a₁, a₂ and a₃

y=M(t,a)+ε=a ₁ +a ₂ cos(2t+a ₃)+ε,

where y=L_(s) is an inductance measured in virtual coordinates, t=λ isan angle in the virtual coordinates, i.e. a direction of measuring inthe stationary coordinates, ε is a term for measuring error and a=[a₁ a₂a₃]^(T)=[L_(s0) L_(s2) φ]^(T) is a parameter vector of the model.

To solve the parameters of the model, it is possible to use a knownconjugate gradient method. However, implementation of the method bycommonly employed DSP processors is difficult, so it is worth whileutilizing the knowledge that${L_{s0} = \frac{L_{sd} + L_{sq}}{2}},{L_{s2} = \frac{L_{sd} - L_{sq}}{2}},$

where inductances L_(sd) and L_(qs) are the direct-axis and thequadrature-axis transient inductances.

The inductance model is simplified by assuming that the transientinductances L_(sd) and L_(sq) are previously known. The transientinductances can be measured when introducing the machine by turning therotor first in a direct-axis position supplying direct current to thestator. After the rotor is turned in the direct-axis position, thestator is supplied in the direct-axis and the quadrature-axis directionswith step-like voltage pulses that cause currents on the basis of whichthe inductances are calculated. Another option is to use the startingprocedure such that, instead of solving the angle giving the minimumerror, parameters L_(s0) and L_(s2) giving the minimum error are solved.

A simplified inductance model is given by

y=M(t,a)+ε=L _(s0) +L _(s2) cos(2t+a)+ε.

Only parameter a remains to be solved, and consequently the algorithmused for the solution can be simplified considerably. Theabove-mentioned known gradient needed for the solution is thensimplified to a common derivative according to the following equation${{\nabla{F(a)}} = {{\frac{}{a}\left\{ {\sum\limits_{i = 1}^{m}\left\lbrack {f_{i}(a)} \right\rbrack^{2}} \right\}} = {2{\sum\limits_{i = 1}^{m}{\frac{{f_{i}(a)}}{a}{f_{i}(a)}}}}}},$

where${\frac{{f_{i}(a)}}{a} = {{- \frac{{M\left( {t_{i},a} \right)}}{a}} = {L_{s2}\sin \quad \left( {{2t_{i}} + a} \right)}}},$

ƒ_(i)=y_(i)−M(t_(i),a) being the difference of the model M and data(t_(i),y_(i)) and m being the number of data.

Minimization of a function with one variable is concerned, which can beimplemented simply by a processor, for instance, by reducing the valueof the target function M with a given step and by selecting a whichproduces the minimum value. In said simplified case, the a producing theminimum value is the solution to the initial angle, i.e. the informationon the angular position of the synchronous machine rotor.

Since the stator inductance is a function of twice the rotor angle, theabove solution is only a candidate for a rotor angle. It is alsopossible that the solution is said candidate plus 180°. Thus it is notknown for sure, whether the direction found is the north or the southpole of rotor magnetization or the permanent magnet.

To find out the polarity, the invention utilizes the fact that the rotormagnetization saturates the direct-axis inductance. On countermagnetization with a stator current, flux density decreases andsaturation decreases, on forward magnetization vice versa. Thus, thedirect-axis inductance has different magnitudes in the direction of 0°and 180°. After finding a solution candidate, the polarity of the rotormagnetization can be found out by measuring the inductance once more inthe direction 8 of the solution candidate and in the direction 10 of180° therefrom. The inductances of both directions having been measured9, the lower one of these inductances is selected 11, 12, 13 to be thecorrect rotor angle.

If the synchronous machine used is a synchronous reluctance machine 7,there is no need to find out the polarity of the machine poles due tothe structure of the machine.

The above-described inductance models do not take the saturation of thedirect-axis inductance into account, so the polarity of the permanentmagnet has to be found out separately. However, it is possible to takethe polarity into account in the model, as a result of which it ispossible to find out the position angle of the rotor without anyseparate step of polarity checking. By assuming that the direct-axisinductance changes linearly as a function of the direct-axis current, itis possible to write for the direct-axis inductance

L _(sd) =L _(sd0) +k·i _(sd)

where k is the slope of the inductance and L_(sd0) is its value, wheni_(sd)=0. By supplying current into the stator in the x-axis directionof the stator coordinates, whereby i_(sy)=0, said current in the rotorcoordinates is given by

i _(sd) =i _(sd) cosθ_(r).

The stator inductance is then $\begin{matrix}{L_{s} = \quad {\frac{L_{sd} + L_{sq}}{2} + {\frac{L_{sd} - L_{sq}}{2}\cos \quad 2\quad \theta_{r}}}} \\{= \quad {\frac{L_{sd0} + L_{sq}}{2} + {\frac{L_{sd0} - L_{sq}}{2}\cos \quad 2\quad \theta_{r}} + {\frac{k}{2}i_{sx}\cos \quad \theta_{r}} + {\frac{k}{2}i_{sx}\cos \quad \theta_{r}\cos \quad 2\quad {\theta_{r}.}}}}\end{matrix}$

In the above-described manner, a replacement θ_(r)→λ−θ_(r) is made,which results in an improved inductance model taking the saturation intoaccount

y=M(t,a)+ε=a₁ +a ₂ cos(t+a ₄/3)+(a₃ +a ₂ cos(t+a ₄/3))cos(2t+a ₃)+ε,

where parameters a_(n) are $a = {\begin{bmatrix}a_{1} & a_{2} & a_{3} & a_{4}\end{bmatrix}^{T} = {\begin{bmatrix}\frac{L_{sd0} + L_{sq}}{2} & {\frac{k}{2}i_{sx}} & \frac{L_{sd0} - L_{sq}}{2} & \phi\end{bmatrix}^{T}.}}$

Depending on the number of previously known parameters, the solution canbe simplified in the above-described manner.

When the direction of the rotor is found out in accordance with theinvention, flux linkages are initialized 14 according to the rotordirection. The objective of the initialization is to adapt the invertercontrol systems to the conditions in the synchronous machine. Afterinitialization, the synchronous machine can be started in a reliablemanner by utilizing any known open-loop vector control method.

It is obvious to a person skilled in the art that as technologyprogresses the basic idea of the invention can be implemented in avariety of ways. Thus, the invention and its embosdiments are notrestricted to the above-described examples but they may vary within thescope of the claims.

What is claimed is:
 1. A method of starting open-loop vector control ina synchronous machine, the method comprising the steps of: determining astator inductance model of the synchronous machine, measuring the statorinductance in a plurality of directions, arranging measured statorinductances in the determined stator inductance model in order to formmodel parameters giving the minimum error, checking the magnetizationpolarity of a rotor in order to verify the direction of the rotormagnetization, initializing flux linkages of the open-loop vectorcontrol according to the formed model parameters and the direction ofthe rotor magnetization, and starting the synchronous machine by thevector control method.
 2. A method as claimed in claim 1, wherein themeasuring of the stator inductance of the synchronous machine in aplurality of directions comprises the steps of: supplying a voltagepulse to the stator of the synchronous machine in a plurality ofdifferent directions, calculating, on the basis of the voltage pulsessupplied in different directions, of the currents generated by thevoltage pulses and of the stator resistance, the magnitudes of the fluxlinkages of corresponding directions, and calculating from themagnitudes of the flux linkages and the currents, the stator inductancesof corresponding directions.
 3. A method as claimed in claim 1, whereinarranging the measured stator inductances in the determined statorinductance model comprises the step of: arranging the measured statorinductances in the stator inductance model by the method of least meansquares so as to provide model parameters, the model parameterscomprising a solution to an initial angle of the rotor.
 4. A method asclaimed in claim 1, wherein checking the rotor magnetization polaritiescomprises the steps of: measuring the stator inductance in the directionof the initial rotor angle solution including model parameters givingthe minimum error, and in the direction of 180 electrical degrees fromthe initial angle solution, and selecting an angle having the lowerstator inductance of the measured inductances to be